What do people think about sharing personal thoughts and feelings online? How secure would the site need to be?

I would expect it to be Fort Knox, with only my therapist getting any information I might put out there.

If I am expected to open up online, about what would be very personal, private issues, I must have no doubt that I can trust the service

And would the content created during the session be stored?

I'm leaning towards yes.... even if it's only for a short while.

I actually do online counseling and privacy is the #1 concern.  Whether you are in an office with a therapist or in your own house, it is important always to remember about your surroundings as well as the site's privacy.

You say you do online now? How did you decide that you could trust them?